When $P$ is constant, it is called type-(v) P-properly $quasi\infty nvex$ in [9]. â150â. Page 3. It should be mentioned that a right $P_{y}$ -convex set-valued map is not nec- .... assumption (AO), that $\varphi(x, y)\not\leqq K(y)$ ; thus $y\in F(x)$ . ..... Let $y\in C$ be fixed and set $y_{t}=ty+(1-t)\overline{x}$ for $t\in[0,1]$ , which ...
Nihonkai Math. J. Vol.12(2001), 149-164
On vector equilibrium problems: Remarks on a general existence theorem and applications El Mostafa Kalmoun, Hassan Riahi and Tamaki Tanaka Abstract
We focus our attention on generalized vector equilibrium problems. In particular, we formulate a general and unified existence theorem, present an analysis for the assumptions used in this result, and give some applications to vector variational inequalities, vector complementarity problems and vector optimization. 2000 Mathematics Subject Classiflcation. $49J35,54H25$ Key words and phrases. Vector equilibrium problems, vector variational$m$ zalike inequalities, vector complementarity problems, vector opt semicontinuity. vector convexity, set-valued transfer tion,
set-valued
Introduction and preliminaries
1
vector In the present paper, we investigate the folowing form of generalized equilibrium problems: Find
$\overline{x}\in C$
such as to satisfy
$\varphi(\overline{x},y)\not\leqq K$
(hi) for all
where $\bullet$
$\bullet$
are topological vector spaces,
$X$
and
$C$
is a nonempty closed
$\bullet$
$\varphi$
:
$E$
$C\times C\rightarrow 2^{E}$
convex subset of $X$ ,
is a set-valued map, and
–149–
$y\in C$
,
(GVEP)
$\bullet$
$K$
is a set-valued map from
$C$
to
$E$
.
In particular, we formulate a general existence result, present an analysis for the assumptions used, and give some applications. Let us first recall and introduce some concepts about convexity and semithe set of all finite continuity for set-valued vector maps. We denote by subsets of $C$ . $\mathcal{F}(C)$
1.1
Generalized set-valued convexity
Deflnition 1.1. We say that $\psi;C\times X\rightarrow 2^{E}$ is diagonally quasi convex in its first argument relatively to , in short L-diagonally quasi $\omega nvex$ in , if and any in $co(A)$ , we have $\psi(A,y)\not\leqq L(y)$ . for any $A$ in $L$
$\mathcal{F}(C)$
$x$
$y$
Let us consider the scalar case, that is when $E=R$ and is a numeric $:=$ ] [) for all $x\in C$ , [ (resp. valued function. If we set $:=$ ] , then the L-diagonal quasi convexity of in colapses to the where following: $\psi$
$L$
$L$
$-\infty,\gamma$
$\gamma\in\overline{R}$
$\gamma,+\infty$
$x$
$\psi$
$\forall A\in \mathcal{F}(C)\forall y\in co(A)$
:
$\dot{m}n\psi(x,y)x\in A\leq\gamma$
(resp.
$\max_{x\in A}\psi(x,y)\geq\gamma$
),
which is the -diagonal quasi convexity (resp. concavity) considered by Zhou and Chen in [20]. Let us also recal the definition of some concepts of convexity for setvalued maps related to moving cones in $E$ . Let $(P(y))_{y\in C}$ be a family of proper for all $y\in C$ . in $E$ with int $\gamma$
$ P(y)\neq\emptyset$
$\infty nes$
Definition 1.2. [$J4/\psi$ : $x_{1},x_{2},y\in C$
and all
$C\times C\rightarrow 2\dot{u}$
$\lambda\in[0,1]$
said to be right
$P_{y}$
-convex if, for all
, one has
$\psi(\lambda x_{1}+(1-\lambda)x_{2}, y)\subseteq\lambda\psi(x_{1}, y)+(1-\lambda)\psi(x_{2}, y)-P(y)$
is said to be right Deflnition 1.3. [ $C$ and all , one has either $3J\psi$
$P_{y}$
-quasiconvexl if, for all
.
$x_{1},$
$ x_{2},y\in$
$\lambda\in[0,1]$
$\psi(\lambda x_{1}+(1-\lambda)x_{2}, y)\subseteq\psi(x_{1}, y)-P(y)$ $or$
$\psi(\lambda x_{1}+(1-\lambda)x_{2}, y)\subseteq\psi(x_{2}, y)-P(y)$
.
1In [3], it is called -quasiconvex-like, and the left -quasiconvexity below is called -quasiconvex in [13]. When $P$ is constant, it is called type-(v) P-properly $quasi\infty nvex$ in [9]. $P_{y}$
$P_{y}$
$P_{y}$
–150–
It should be mentioned that a right -convex set-valued map is not necessarily right -quasiconvex. When $F$ is single-valued and $P$ is constant, we recover the P-convexity and the properly quasi-P-convexity of [8], respectively. All these concepts reduce to the ordinary convexity and quasiconvexity when considering the scalar case. Furthermore, Konnov and Yao [13] work with following set-valued concept. $P_{y}$
$P_{y}$
is said to be Definition 1.4. $C$ and all $\lambda\in[0,1]$ , one has either $\int 13J\psi$
left
$P_{y}$
for all
-quasiconvex if,
$x_{1},$ $x_{2},$
$ y\in$
$\psi(x_{1},y)\subseteq\psi(\lambda x_{1}+(1-\lambda)x_{2},y)+P(y)$ $\sigma r$
$\psi(x_{2}, y)\subseteq\psi(\lambda x_{1}+(1-\lambda)x_{2},y)+P(y)$
1.2
.
Transfer semicontinuity
Deflnition 1.5. We say that is K-transfer semicontinuous in $(x,y)\in C\times X$ with $\varphi(x,y)\subset K(y)$ , there exist an element open $V\subset X$ containing such that for all $v\in V$ . $y$
$\varphi$
if for any
$x^{\prime}\in C$
and an
$\varphi(x^{\prime},v)\subset K(v)$
$y$
An equivalent statement is as follows: for each $\forall x\in C\exists y_{\alpha}\rightarrow y$
in
$X$
:
$y\in X$
satisfying
$\varphi(x,y_{\alpha})\not\leqq K(y_{\alpha})$
,
we have $\varphi(x,y)\not\leqq K(y)$
$\forall x\in C$
.
is said to be transfer $u.s.c$ . (resp. $l.s.c.$ ) in Definition 1.6. $(x,y)\in C\times X$ and any open $N\subset E$ with $\varphi(x,y)\subset N$ (resp. and an open $V\subset X$ such that there exist an element ) for all $v\in V$ . (resp. $\varphi$
$y$
if,
for any
$\varphi(x,y)\cap N\neq\emptyset$
$x^{\prime}\in C$
)
$\varphi(x^{\prime},v)\subset N$
$\varphi(x^{l},v)\cap N\neq\emptyset$
Note here that an u.s. . (resp. l.s. ) set-valued bi-map in one of its arguments is obviously transfer u.s. . (resp. l.s. ). Moreover, the relationship between the two concepts of transfer continuity, introduced above, is the following. $c.$
$c$
$c.$
$c$
Proposition 1.1. If is transfer $u.s.c$ . in with compact values and has an open graph, then is K-transfer semicontinuous in . $y$
$\varphi$
$y$
$\varphi$
–151–
if $K$
Proof.
Let
such that for any $x\in C$ there is a net converging to in with . Therefore, for every $x\in C$ , there exists . Passing to a subnet, if necessarily, we can with certainly assume that, for every $x\in C$ there exists $z\in\varphi(x, y)$ such that . Otherwise, there exists such that, for every , there is an open $N_{z}\subset E$ containing and satisfying, for every in $X$ and for every for sufficiently large. Since and is compact, : for some finite subset in . Hence, there exists such $X$ that, for every in and for large enough, because of the transfer upper semicontinuity of . But, we can also write, for some in $X$ and some for large enough. It follows $X$ that for some in , which is a contradiction. The same argument is still applicable to show the assertion on compact subsets of $X$ . This completes our proof. $X$
$y$
$y\in X$
$(y_{\alpha})$
$\varphi(x, y_{\alpha})\not\leqq K(y_{\alpha})$
$z_{\alpha}\in\varphi(x, y_{\alpha})$
$z_{\alpha}\not\in K(y_{\alpha})$
$x^{\prime}\in C$
$z_{\alpha}\rightarrow z$
$z\in\varphi(x^{\prime}, y)$
$z$
$z_{\alpha}^{l}\in\varphi(x^{\prime}, y_{\alpha}^{\prime}),$
$y_{\alpha}^{\prime}\rightarrow y$
$z_{\alpha}^{\prime}\not\in N_{z}$
$\varphi(x^{j},y)$
$\varphi(x^{\prime}, y)\subset\bigcup_{z\in\varphi(x^{l},y)}N_{z}$
$\{z_{1}, \ldots z_{n}\}$
$\alpha$
$\varphi(x^{l},y)\subset N=\bigcup_{1=1}^{1=n}N_{z}$ $x^{\prime\prime}\in C$
$\varphi(x^{\prime}, y)$
$y_{\alpha}^{\prime}\rightarrow y$
$\varphi(x^{j\prime},y_{\alpha}^{\prime})\subset N$
$\alpha$
$\varphi$
$z_{\alpha}\in\varphi(x^{\prime\prime},y_{\alpha}),$
$y_{\alpha}\rightarrow y$
$\varphi(x^{\prime\prime}, y_{\alpha})\not\leqq N$
$z_{\alpha}\not\in N$
$\alpha$
$y_{\alpha}\rightarrow y$
$\blacksquare$
2
The existence theorem
For the proof of the main result, we need the folowing KKM lemma, which is a particular case of of the extended version of Fan-KKM theorem [11, Theorem 2.1].
Lemma 2.1. Assume that $(Oi)G(x)\subset F(x)$
$F,$ $G:C\rightarrow 2^{X}$
for all $x\in C$ ,
(i)
$G$
is a KKM map,
(ii)
$F$
is
transfer closed-valued,
(iii) there is a compact subset exists a compact convex
$B$
$B_{A}$
in $X$ such that for each $A\in \mathcal{F}(C)$ there in $X$ contain ing $A$ such that
$\cap$
$\overline{G(x)}\cap B_{A}\subset B$
$x\in B_{A}\ulcorner C$
Then
satisfy
$\bigcap_{x\in C}B\cap F(x)\neq\emptyset$
.
–152–
.
Recall that $G$ is a KKM map when $co(A)\subset\bigcup_{x\in A}G(x)$ for any subset , and that $F$ is transfer closed-valued whenever $y\in X$ and $x\in C$ . such that such that $y\not\in F(x)$ there exists Now we are in position to state the following existence theorem for (GVEP) in topological vector spaces. $A\in \mathcal{F}(C)$
$x^{\prime}\in C$
$y\not\in\overline{F(x^{\prime})}$
Theorem 2.1. Suppose that (A )
$\psi(x, y)\not\leqq L(y)\Rightarrow\varphi(x, y)\not\leqq K(y)\forall x,$
(A 1)
$\psi$
$0$
$(A2)\varphi$
is L-diagonally quasi-convex in
is
K-tmnsfer semicontinuous
$x$
$y\in C$
;
;
in ; $y$
(A 3) there is a nonempty compact subset $B$ in $X$ such that for each $ A\in$ there is a compact convex $B_{A}\subset X$ containin $A$ such that, for every , there exists $x\in B_{A}\cap C$ with $\mathcal{F}(C)$
$g$
$y\in B_{A}\backslash B$
$y\in int_{X}\{v\in X : \psi(x, v)\subseteq L(v)\}$
Then there enists
Proof.
$\overline{y}\in B$
such that
.
for all $x\in C$ . $G:C=X$ as follows:
$\varphi(x,\overline{y})\not\leqq K(\overline{y})$
Define two set-valued maps
$F,$
$F(x)=\{y\in X : \varphi(x, y)\not\leqq K(y)\}$
and $G(x)=\{y\in X : \psi(x, y)\not\leqq L(y)\}$
The existence of a generalized vector equilibrium for $K$ is now equivalent to
.
$\varphi$
in
$B$
with respect to
$\bigcap_{x\in C}B\cap F(x)\neq\emptyset$
Hence, we need only to check the assumptions of Lemma 2.1 for
$F$
and
$G$
.
(Oi) Let $x\in C$ and let $y\in G(x)$ ; then $\psi(x, y)\not\leqq L(y)$ ; it follows, by assumption (AO), that $\varphi(x, y)\not\leqq K(y)$ ; thus $y\in F(x)$ . and let $y\in coA$ . By assumption (A1), we have $L(y)$ . Hence there exists $x\in A$ such that $\psi(x, y)\not\leqq L(y)$ ; that is $y\in G(x)$ . Hence $co(A)\subset\bigcup_{x\in A}G(x)$ . We conclude that $G$ is a KKM
(i) Let
$\psi(A, y)\not\leqq$
$A\in \mathcal{F}(C)$
map.
–153–
(ii) Let us consider $(x, y)\in C\times X$ with $y\not\in F(x)$ . Suppose, contrary to our claim, that $y\in cl_{X}F(x^{\prime})$ for al $x^{l}\in C$ . Therefore, for every , $X$ there is a net in converging to and satisfying . It folows that for al since is Ktransfer semicontinuous in , and hence that , for all $F$ a contradiction. We conclude that is transfer closed-valued. $x^{\prime}\in C$
$(y_{\alpha})$
$K(y_{\alpha})$
$\varphi(x^{\prime}, y_{\alpha})\not\leqq$
$y$
$x^{\prime}\in C$
$\varphi(x^{\prime},y)\not\leqq K(y)$
$\varphi$
$y\in F(x^{\prime})$
$y$
$x^{\prime}\in C$
(iii) It suffices to see that assumption (A3) leads to $B_{A}\backslash B\subset\bigcup_{x\in B_{A}\cap C}int_{X}G(x)$
hence
$ B_{A}\cap\bigcap_{x\in B_{A}r\kappa)}cl_{X}G(x)\neq\emptyset$
;
.
The proof is complete.
$\blacksquare$
Theorem 2.1 generalizes [3, Theorem 2.1], which is proved by means of a Fan-Browder fixed point theorem- an immediate $\infty naequence$ of the FanKKM theorem. As we wil mention in the ‘Assumptions analysis’ subsection, our hypotheses are more general than those used in [3]. Besides, the scalar version of this result extends [17, Theorem 4] (we take $C_{A}=co(A\cup R)\cap X$ where $R$ is the convex compact which contains $C$ in [17, Theorem 4, ]). Other particular cases are [1, Theorem 2], [19, Theorem 2.1], [20, Theorem 2.11], [18, Theorem 1], [15, Corolary 2.4], [16, Lemma 2.1] and [4, Theorem 2]. The origin of this kind of results goes back to Ky Fan [7]. His classical minimax inequality can be deduced from our result by setting $E=R,$ $K(x)=$ and $\varphi(x,y)=\psi(x,y)=f(x,y)-\sup_{x\in}f(x,x)$ for all $x,y\in C$ . $(4i\ddot{u})$
$R_{+}^{*}$
3
Assumptions analysis
In this section, we analyze the requirements of Theorem 2.1 by presenting different situations where assumptions $(AO)-(A3)$ hold true. Let $(P(y))_{y\in C}$ a family of proper convex closed cones on $E$ with int for al $y\in C$ . $ P(y)\neq\emptyset$
$\bullet$
Pseudomonotonicity
Remark 3.1. (A ) holds provided one $0$
satisfied.
–154–
of the folloutng statements
is
$(a)\varphi=\psi$
and $K=L$ .
$(b)X=C,$ $K(y)=-L(y)=-int$ $P(y),$ $x,$
$y\in C$
, and
$\varphi$
is
$P_{x}$
$\psi(x,y)=\varphi(y,x)$
$\varphi(x, y)\not\leqq intP(x)\Rightarrow\varphi(y,x)\not\leqq-int$ $P(x)\forall x,$
$\bullet$
Remark 3.2. (A 1) holds provided that, for every $y\in $(a)\psi(y,y)\not\leqq L(y)$
the set
all
$y\in C$
.
Suppose that $X=C$ .
Convexity.
$(b)$
for
-pseudomonotone, that is,
C$
, one has either
, and
$\{x\in C:\psi(x, y)\subseteq L(y)\}$
is convex,
$or$
(i) $L(y)=-int$ $P(y)$ and
(ii) (iii)
$\psi(y,y)\subseteq P(y)$
$\psi$
is
left
$P_{\nu}$
is w-p ointed 2
$P(y)$
, and
-quasiconvex.
, $x_{n}\in C$ , Indeed, if we suppose in contrary that there exist such that $\sum_{i=1}^{n}k=1$ and , and $x_{1},$
$\lambda_{1)}\ldots\lambda_{n}\in[0,1]$
$\ldots$
$y=\sum_{j=1}^{n}\lambda_{j}x_{j}$
(1)
$\psi(x_{i},y)\subseteq L(y)$
for each $i=1,$ $\ldots n$ . For the first assertion, assumption (b) shows that $\psi(y,y)\subseteq L(y)$ , which contradicts (a). For the second one, (iii) implies $\psi(x_{i_{0}},y)\subseteq\psi(y, y)+P(y)\subseteq P(y)+P(y)\subseteq P(y)$
for some $i_{0}\in\{1, \ldots n\}$ ; this contradicts (1) since $P(y)$ is w-pointed. The assertion is proved. Moreover condition (b) in Remark 3.2 is satisfied provided that $L(y)$ is $x_{2}\in C$ and all $\lambda\in[0,1]$ , convex, and for all $x_{1},$
$\psi(\lambda x_{1}+(1-\lambda)x_{2},y)\subseteq\lambda\psi(x_{1}, y)+(1-\lambda)\psi(x_{2}, y)$
In case of $L(y)=-int$ $P(y),$
2A cone
$P$
is w-pointed if $P\cap-
int$
holds if either
$(b)$
$ P=\emptyset$
.
–155–
,
$\psi$
-
$\bullet$
$\psi$
is right P-convex, or is right P-quasiconvex.
Continuity
Remark 3.3. is
$(A8)$
holalS provided that one
satisfied. is (tmnsfer) open gmph.
$(a)\varphi$
$(b)\varphi$
$\dot{u}$
$(c)$
$u.s.c$
. in
is (tmnsfer) $u.s.c$ . in an open subset of $E$ .
For each
$x\in C$
, the set
$y$
$y$
of the following statement
utth compact values and
and $K(x)=O$
if $K$
for all $x\in C$ ,
$\{y\in X:\varphi(x,y)\not\leqq K(y)\}$
has an
where
is closed in
$O$
$C$
.
While the assertion $(c)\Rightarrow(A2)$ is obvious, the other assertions follow ffom Proposition 1.1. $\bullet$
Coercivity. Recal first the following definition. Definition 3.1. We will say that a set-valued map : K-compact if the set $cl\{y\in D:\phi(y)\not\leqq K(y)\}$ is compact in Y. $\phi$
$D\rightarrow 2^{E}$
is
It has to be observed that this definition is equivalent to the following fact: there exists a nonempty compact subset $B$ in $D$ such that $K(y)$ for al . $\phi(y)\subseteq$
$y\in D\backslash B$
Remark 3.4. $(a)X$ $(b)$
$(c)$
holds
if one of the folloutng statements is satisfied.
is compact.
There is $x_{0}\in C$ such that is K-compact. There is a nonempty compact subset $B$ in $X$ such that $\psi(x_{0}, \cdot)$
$y\in X\backslash B$
$(d)$
$(A3)$
there enists
$x\in B\cap C$
such that
There is a nonempty compact subset $B$ subset such that for each $B^{\prime}\in X$
of
$X$
$y\in X\backslash B$
each
.
and a compact convex there exists $x\in B^{\prime}\cap C$
with
$y\in int_{X}\{v\in X : \psi(x,v)\subseteq L(v)\}$
–156–
for
$\psi(x, y)\subseteq L(y)$
.
Besides, when the classical assumption (A 3) holds provided that $(e)$
$(c)$
of Remark 3.3
is satisfied,
there is a nonempty compact subset $B$ in $X$ such that for each there is a compact convex $B_{A}\subset X$ containing $A$ such that, for every $y\in C\backslash B$ , there exists $x\in B_{A}\cap C$ with
$A\in \mathcal{F}(C)$
$\varphi(x, y)\subseteq$
$K(y)$
4
.
Applications
Throughout this section, and otherwise stated, $C$ is supposed to be a nonempty closed convex subset in a t.v. $sX$ , and $E$ to be a real topological vector space. Assume also we are given a family $\{P(x) : x\in C\}$ of convex cones in $E$ with int for all $x\in C$ . Also, we denote by $L(X, E)$ the space of all linear $ P(x)\neq\emptyset$
bounded applications from
4.1
$X$
to
$E$
.
Generalized vector variational like-inequalities
Let us consider a set-valued operator $T$ from $C$ into $L(X, E)$ , and a bifunction : from $C$ to itself. We write for $\Pi\subset L(X, E)$ and $x\in C,$ $\pi\in\Pi\}$ , where denotes the evaluation of the linear mapping at which is supposed to be continuous on $L(X, E)\times X^{3}$ . The generalized vector variational inequality problem (GVVLIP) takes the following form: $\langle\Pi, x\rangle=\{\langle\pi, x\rangle$
$\eta$
$\pi$
$\langle\pi,x\rangle$
Find
$\overline{x}\in C$
such that,
$\langle T\overline{x}, \eta(y,\overline{x})\rangle\not\subset-int$
$P(\overline{x})\forall y\in C$
.
Thus (GVVLIP) is a particular case of (GVEP) if we take $\varphi(x,y)=\{\langle t, \eta(y, x)\rangle :
t\in Tx\}$
.
For the reader’s convenience, we recall the folowing definitions. Deflnition 4.1. (see $[2J$) 1) $T$ is said to be -pseudomonotone if, $\eta$
for all
$x,$
$y\in C$
,
$\langle Tx, \eta(y, x)\rangle\not\subset-int$ $P(x)\Rightarrow\langle Ty, \eta(y, x)\rangle\not\subset-int$
3A typical situation when
$X$
is a reflexive Banach and
–157–
$E$
$P(x)$
is a Banach
.
$x$
2)
$T$
is said to be V-hemicontinuous iffor any $y\in C$ and ] $0,1[$ $T(tx+$ (that is, for any $z_{t}\in T(tx+(1-t)y)$ there exists as Ty$ such that $a\in C,$ any ). as for
$z\in
$ t\in$
$x,$
$(1-t)y)\rightarrow T(y)$
$t\rightarrow 0_{+}$
$\langle z_{t}, a\rangle\rightarrow\langle z, a\rangle$
$t\rightarrow 0_{+}$
It has to be observed that when $T$ is single-valued, we recover the hemicontinuity used in [5]. If $\eta(x,y)=x-y$ for all $x,y\in C,$ is dropped from the definition of pseudomonotonicity. The linearization lemma plays a significant role in variational inequalities. Chen [5] extended this lemma to the single-valued vector case. For our need in this subsection, we state it in the set-valued case by using standard Minty’s argument. Consider the following problem, which may be seen as a dual problem of (GVVLIP), $\eta$
Find
$\overline{x}\in C$
such that
$\langle Ty,\eta(y,\overline{x})\rangle\not\subset-int$ $P(\overline{x})\forall y\in C$
.
$(GVVLIP^{*})$
Lemma 4.1. Suppose that affine and $\eta(x,x)=0$ for each $ x\in$ $T$ C. If is -pseudomonotone and V-hemicontinuous then (GVVLIP) and $(GVVLIP*)$ are equivalent. $\eta(\cdot,x)\dot{u}$
$\eta$
Proof. We only need to show that every solution of $(GVVLIP*)$ is a solution of (GVVLIP); the reverse assertion folows clearly from the pseudomonotonicity of $T$ . To this end, let such that $\eta-$
$\overline{x}\in C$
$\langle Ty,\eta(y,\overline{x})\rangle\not\subset-int$
Let
be fixed and set convex subset $C$ . Then $y\in C$
$P(b1)\forall y\in C$
$y_{t}=ty+(1-t)\overline{x}$
$\langle Ty_{t},\eta(y_{t},\overline{x})\rangle\not\subset$
for
.
$t\in[0,1]$
, which is in the
-int $P(b1)$ .
Hence $\langle Ty_{t},\eta(y,\overline{x})\rangle\not\subset-int$
$P(\overline{x})$
By the V-hemicontinuity of $T$ and the closedness of that $\langle T\overline{x},\eta(y,\overline{x})\rangle\not\subset-int$
The last assertion holds for an arbitrary
$P(\overline{x})$
$y\in C$
–158–
. $E\backslash $
(-int
$P(\overline{x})$
), it follows
.
. The lemma is proved.
$\blacksquare$
As an application of Theorem 2.1, we are now in position to formulate the following existence result for (GVVLIP). Theorem 4.1. Suppose that (i) the mapping int
(ii)
for each $x\in C,$
(iii)
$T$
$P(\cdot)$
has an open graph in
$\eta(\cdot, x)$
is compact valued,
$\eta$
is affne,
$C\times L(X, E)$
;
is continuous and
$\eta(x, \cdot)$
$\eta(x, x)=0$
;
-pseudomonotone and V-hemicontinuous;
(iv) there is a nonempty compact subset $B$ in $C$ such that for each $ A\in$ there is a compact convex $B_{A}\subset C$ containing $A$ such that, for every , there enists $x\in B_{A}\cap C$ utth $\mathcal{F}(C)$
$y\in B_{A}\backslash B$
$y\in int_{C}\{v\in C:\langle Tv,\eta(x,v)\rangle\subseteq-int P(v)\}$
Then (GVVLIP) has at least one solution, which is in
$B$
.
.
Set $\varphi(x,y)=\langle Tx,\eta(x,y)\rangle,$ $\psi(x,y)=\langle Ty,\eta(x,y)\rangle$ and $K(x)=$ -int $P(x)$ for al $x,y\in C$ . Let us show that the assumptions of Theorem 2.1 are satisfied: (AO) follows clearly from the -pseudomonotonicity of $T$ . (A1) First we have, $\psi(x, x)=\langle Tx,\eta(x,x)\rangle=0\not\in-int$ $P(x)$ for all $x\in C$ , and for being fixed in $C$ , the set $P$
rvof.
$\eta$
$\bullet$
$\bullet$
$y$
$\{x\in C:\langle Ty,\eta(x,y)\rangle\subseteq-intP(y)\}$
is convex since is affine. Thus conditions (a) and (b) of Remark 3.2 hold, which lead to (A1). (A2) Fix in $C$ and let be a net on $C$ converging to $y\in C$ such that $\eta(\cdot, y)$
$x$
$(y_{\alpha})$
$\langle Tx,\eta(x,y_{\alpha})\rangle\not\subset-int$ $P(y_{\alpha})$
Therefore there exists
$z_{\alpha}\in Tx$
.
such that
$\langle z_{\alpha}, \eta(x, y_{\alpha})\rangle\not\in-int$ $P(y_{\alpha})$
.
is compact then, passing to a subnet, if necessarily, we may assume , and that converges to $z\in Tx$ . By the continuity of the map we get . Hence, according to (i), we obtain
Since
$Tx$
$\eta(x, \cdot)$
$z_{\alpha}$
$\langle z_{\alpha}, \eta(x, y_{\alpha})\rangle\rightarrow\langle z, \eta(x, y)\rangle$
$\langle z, \eta(x,y)\rangle\not\in-int$
$P(y)$
–159–
;
$\langle., .\rangle$
thus $\langle Tx, \eta(x, y)\rangle\not\subset-int$
(A3) follows easily from (iv). Therefore, such that
$P(y)$
$hom$
.
Theorem 2.1, there exists
$\overline{x}\in B$
$\langle Ty, \eta(y,\overline{x})\rangle\not\subset-int$
Hence $(GVVLIP*)$ has a solution in theorem according to Lemma 4.1.
$B$
$P$
(b1)
$\forall y\in C$
.
, which completes the proof of the $\blacksquare$
The coercivity condition (iv) is better than that formulated in [12], namely, there is a compact $B$ of $X$ and $x_{o}\in C\cap B$ such that $(Tx_{o},$
4.2
$\eta(x_{o},y)\rangle$
$\subset-int$ $P(y)\forall y\in C\backslash B$
.
Vector complementarity problems
A natural extension of the classical nonlinear complementarity problem, $(CP)$ for short, is considered as follows. Let $T$ be a single-valued operator from $C$ , which is supposed to be a convex closed cone, to $L(X, E)$ . The vector complementarity problem considered in this subsequent, $(VCP)$ for short, is to find such that $\overline{x}\in C$
$\langle T(\overline{x}),\overline{x}\rangle\not\in intP(\overline{x})$
, and
$\langle T(\overline{x}),y\rangle\not\in-int$
$P(\overline{x})$
for al
$y\in C$
.
This problem collapses to $(CP)$ when $E=R$ and $P(x)=R_{+}$ for al $x\in C$ . By means of vector variational inequalities, we can formulate the following existence theorem for $(VCP)$ . Theorem 4.2. Suppose that (i) the set-valued map int (ii) (iv)
$T$
$P(\cdot)$
has an open graph in
$C\times L(X,E)$
;
is pseudomonotone and hemicontinuous;
there is a nonempty compact subset $B$ in $C$ such that for each $ A\in$ there is a compact convex $B_{A}\subset C$ containing $A$ such that, for every , there $e\dot{m}tsx\in B_{A}\cap C$ utth
$\mathcal{F}(C)$
$y\in B_{A}\backslash B$
$y\in int_{C}\{v\in C:\langle Tv, x-v\rangle\in-int P(v)\}$
–160–
.
Then
$\eta(x, y)=x-y$
. $\overline{x}\in B$
$x,$
$\langle T\overline{x}, z-\overline{x}\rangle\not\in-int$
Since
$B$
It is clear that all the assumptions of Theorem 4.1 are satisfied such that for all $y\in C$ . Therefore there exists
Proof. with
has at least one solution, which is in
$(VCP)$
$P(\overline{x})\forall z\in C$
(2)
.
is a convex cone, then setting in (2), $z=0$ and arbitrary $y\in C$ , we get respectively $C$
and
$\langle T\overline{x},\overline{x}\rangle\not\in intP(\overline{x})$
$\langle T\overline{x}, y\rangle\not\in-int$ $P(\overline{x})$
Hence we conclude that hi is also a solution to
4.3
$z=y+\overline{x}$
for an
.
$(VCP)$ .
$\blacksquare$
Vector optimization
Here, to convey an idea about the use of vector variational-like inequalities in vector optimization which involves smooth vector mappings, we prove the existence of solutions to weak vector optimization problems, (WVOP) for short, by considering the concept of invexity. Let us state the problem as follows. Find
$\overline{x}\in C$
such that
$\phi(y)-\phi(\overline{x})\not\in-int$ $P$
for all
(WVOP)
$y\in C$ ,
where $\phi:C\rightarrow E$ be a given vector-valued function and $P$ is a given convex cone in $E$ . the R\’echet Let : $C\times C\rightarrow X$ be a given function, and denote by derivative of once the latter is assumed to be R\’echet differentiable. $\nabla\phi$
$\eta$
$\phi$
Definition 4.2. Suppose that P-in vex uri th respect to if
$\phi$
is Fk\’echet
differentiable.
$\phi$
is said to be
$\eta$
$\phi(y)-\phi(x)-\langle\nabla\phi(x), \eta(y, x)\rangle\in P$
$\forall x,y\in C$
.
If $E=R$ and $P=R^{+}$ then we recover the real invexity introduced by Hanson [10] and later labeled so by Craven [6] due to its “ invariance” under convex” transformations.
“
Theorem 4.3. Suppose that $P$ is a convex cone in $E$ with let : be a Fbl\’echet differentiable mapping. Assume that
$ intP\neq\emptyset$
$\phi$
$C\rightarrow E$
–161–
, and
(i) (ii)
(iii)
(iv)
$P$ $x,y\in C\langle\nabla\phi(x), \eta(y, x)\rangle\not\in-int$
$\phi$
implies (
$\not\in-intP$
$\nabla\phi(y),\eta(y, x)\rangle$
for
all
is P-invex utth respect to ; $\eta$
$\nabla\phi\dot{u}$
hemicontinuous;
for each $x\in C,$
$\eta(\cdot, x)$
is affine,
$\eta(x, \cdot)$
is
$\omega ntinuous$
and $\eta(x,
x)=0$
;
(v) there is a compact subset $B$ in $X$ such that for every finite subset $A$ in $C$ there is a compact convex $C_{A}\subset X$ containing $A$ such as to satisfy, for $\in-int$ $P$ . every , there $e\dot{m}tsx\in C_{A}\cap C$ with $y\in C\backslash B$
$(\nabla\phi(x),$ $\eta(x,y)\rangle$
Then (WVOP) has at least one solution.
Proof.
First,
$byvirtueofTh\infty rem4.lwithT:=\nabla\phi$ , $\langle\nabla\phi(\overline{x}),\eta(y,\overline{x})\rangle\not\in-int$
Then the P-invexity of
$\phi$
allows us to
$P\forall y\in C$
$\infty nclude$
we get
.
.
$\blacksquare$
Acknowledgments. The first author would like to appreciate the warm hospitality of Niigata university during his stay between May and October, 2001 at the Graduate School of Scienoe and Technology. Financial support by he Matsumae International Foundation is gratefully acknowledged.
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E. M. Kalmoun and H. Riahi Cadi Ayyad University, Faculty of Science Semlalia, Department matics, B.P. 2390, Mamrakech-40000, Morocco. T. Tanaka Graduate School 2181, Japan.
of Science and
of Mathe-
Technology, Niigata University, Niigata 950-
Received September 21, 2001
–164–
